#6570810

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,30,35,32,28,31,27)(29,33,40,37,39,36,38,34);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,32,35)(26,30,28,31)(29,37,39,34)(33,38,36,40);(1,4,5,3)(2,6,7,8)(10,15)(11,18)(16,22)(19,24)(25,28,32,26)(27,30,35,31)(29,36,39,33)(34,40,37,38);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,33)(27,34)(28,36)(30,40)(31,38)(32,39)(35,37);(12,20)(16,22)(19,24)(21,23)(25,30,32,31)(26,27,28,35)(29,38,39,40)(33,37,36,34)

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478006);